Thanks to Adam's comment pointing me to Gilles' answer, I finally achieved the effect I wanted with the 3M Ergonomic Mouse. I'm now using KDE Plasma 5, but I suspect the following works for most modern X.Org systems.
According to the command:
The 3M Ergonomic Mouse is labelled "PixArt USB Optical Mouse" on my system, and the command
xinput --list-props 'PixArt USB Optical Mouse'
confirmed that the evdev wheel emulation properties exist for this device.
The evdev man page confirms the xorg.conf Option name for each of these properties.
So I created a directory called
/etc/X11/xorg.conf.d/ and then created a file in this directory called
mousewheel-emulation.conf which contained the following:
# Enable scroll-wheel emulation on the 3M Ergenomic Mouse.
Identifier "3M Ergonomic Mouse Wheel Emulation"
MatchProduct "PixArt USB Optical Mouse"
Option "EmulateWheel" "on"
Option "EmulateWheelButton" "2"
Option "XAxisMapping" "6 7"
Option "YAxisMapping" "4 5"
After rebooting the machine I found that holding the third button (the one which runs up the grip) and moving the mouse up/down now emulates vertical scroll wheel movement, whilst holding the third button and moving the mouse left/right emulates horizontal scroll wheel movement. Very pleasing after so long without the ability to scroll in all applications.
As a note of warning: backup your entire system before you make these changes. When I tried to use the configuration exactly as shown in Gilles' answer, it caused my system to boot to a black screen and wouldn't allow me to switch to a non-graphical terminal. After several reboots I was able to switch to a non-graphical terminal (by holding Ctrl+Alt+F1), then login and remove the new configuration file, which removed the problem. But it's best to backup your files just in case.